Study Guide: Introduction

In one sense, mathematics at university follows on directly from school mathematics. In another sense, university mathematics is self-contained and requires no prior knowledge. In reality, neither of these descriptions is anything like complete. Although it would be impossible to study mathematics at Oxford without having studied it before, there is a marked change of style at university, involving abstraction and rigour. As an undergraduate in any subject, your pattern and method of study will differ from your schooldays, and in mathematics you will also have to master new skills such as interpreting mathematical statements correctly and constructing rigorous proofs.

These notes are an introduction to some of these aspects of studying mathematics at Oxford. Their purpose is to introduce you to ideas which you are unlikely to have met at school, and which are not covered in textbooks. It is not expected that you will understand everything on first reading; for instance, some of the examples may be taken from topics which you have not covered at school, and some of the material in Part II needs time to be absorbed. With the exception of a few fleeting references to physical applied mathematics, the notes discuss topics which are common to all the mathematics courses in Oxford, so they should be almost as useful for students in Mathematics & Philosophy and Mathematics & Computation as for those in Mathematics. It is hoped that after a term or two you will have absorbed all the advice in the notes and that they will become redundant. For this reason, there is no attempt to cover any topics which will not arise until later in your course, e.g. examinations and options.

It is important to appreciate that the notes are an introduction to studying mathematics, not an introduction to the mathematics; indeed, there is very little genuine mathematics in the notes. In particular, the notes attempt to answer the question: How to study mathematics? but not the questions: Why study mathematics? or: What topics do we study in university mathematics? Occasionally, the notes consider the question: Why do we do mathematics the way we do? but the emphasis is on How? not Why? or What? Consequently, the notes have some of the style of a manual, and they will not make the most exciting of reads.

You are therefore recommended to seek other sources for the other questions. The Undergraduate Course Handbook is available online, and gives some idea of What? is studied in the undergraduate course.
At present, the best book known to me, to answer Why? is Alice in Numberland, by J. Baylis and R. Haggarty (Macmillan). Indeed, I recommend that you read Alice in Numberland (at least a substantial part of it) before you read these notes (at least before you read Part II). A list of useful resources (other books and online materials) can be found here: Part 2: Concluding Remarks.

Your tutors are likely to offer their own advice which sections of these notes you should read at what time and what else you should read, and to set you some problems. I will therefore refrain from proffering guidance except to say that Part I will be most useful if it is read before your courses start in the first full week after your arrival in Oxford. It should not take long to read this part as it contains very little mathematics.

Part I contains many instructions and imperatives: Do this; Do that; You should ... They are to be regarded as recommendations or advice, rather than rules which must be religiously observed. (Indeed, it is unlikely that anyone would have time or inclination to carry out all the tasks suggested.) Students differ in temperament and intellect, so what suits one undergraduate will not necessarily suit another. It is up to you to work out your own work-style; it is hoped that these notes will assist you.

Part II is much more technical than Part I, and will need to be read carefully. Many of the individual sentences may need to be read several times to understand their precise meaning and significance. Since some of the ideas are sophisticated, there may be some points which remain unclear to you even after hard thought, particularly if you are reading them before the course has started. If so, do not be alarmed-all may become clear when the course starts and you have a context in which to consider the point. You should be able to understand later sections without having grasped every point earlier on. In the chapter concerning proof it may be more profitable if you pick out some of the examples which are at the appropriate level for you than if you work through them all at one reading.